Summary and Analysis

Study Tools

Special Graphs

Terms

Problems

If a graph does not change when reflected over a line or rotated around a point,
the graph is symmetric with respect to that line or point. The following
graph is symmetric with respect to the
x
-axis (
y = 0
). Note that if
(x, y)
is a point on the graph, then
(x, - y)
is also a point on the graph.

Symmetry with Respect to the
x
-axis

If a function is symmetric with respect to the
x
-axis, then
f (x) = - f (x)
.

The following graph is symmetric with respect to the
y
-axis (
x = 0
). Note
that if
(x, y)
is a point on the graph, then
(- x, y)
is also a point on the
graph.

Symmetry with Respect to the
y
-axis

If a function is symmetric with respect to the
y
-axis, then
f (x) = f (- x)
.

If a graph can be reflected over a line without altering the graph, then that
line is called the axis of symmetry. In the following graph,
x = 2
is the
axis of symmetry. Note that if
(2 + x, y)
is a point on the graph, then
(2 - x, y)
is also a point on the graph.

Axis of Symmetry

If a function has an axis of symmetry
x = a
, then
f (x) = f (- x + 2a)
.

The following graph is symmetric with respect to the origin. In other words, it
can be rotated
180
o
around the origin without altering the graph. Note
that if
(x, y)
is a point on the graph, then
(- x, - y)
is also a point on the
graph.

Symmetry with Respect to the Origin

If a function is symmetric with respect to the origin, then
f (x) = - f (- x)
.